For direction cosines $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$ Using identity $\cos 2\theta = 2\cos^2\theta -1$ So $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ $=(2\cos^2\alpha-1)+(2\cos^2\beta-1)+(2\cos^2\gamma-1)$ $=2(\cos^2\alpha+\cos^2\beta+\cos^2\gamma)-3$ $=2(1)-3$ $=-1$